I hold an investment that will pay me every year for 5 years starting next year. The first payment is 100 units, and each payment is expected to grow by 3% each year. If the interest rate is 11%, what is the present value of the investment.
>

$\mathrm{with}\left(\mathrm{Finance}\right)\:$

>

$\mathrm{growingannuity}\left(100\,0.11\,0.03\,5\right)$

This can also be calculated as follows.
The cash flows are given by:
>

$\mathrm{cf}\u2254\left[100\,100\cdot 1.03\,100{1.03}^{2}\,100{1.03}^{3}\,100{1.03}^{4}\right]$

${\mathrm{cf}}{:=}\left[{100}{\,}{103.00}{\,}{106.0900}{\,}{109.272700}{\,}{112.5508810}\right]$
 (2) 
or equivalently as
>

$i\u2254\'i\'\:$

>

$\mathrm{cf}\u2254\left[\mathrm{seq}\left(\mathrm{futurevalue}\left(100\,0.03\,i\right)\,i\=0..4\right)\right]$

${\mathrm{cf}}{:=}\left[{100.0}{\,}{103.00}{\,}{106.0900}{\,}{109.272700}{\,}{112.5508810}\right]$
 (3) 
>

$\mathrm{cashflows}\left(\mathrm{cf}\,0.11\right)$

Here, we deal with a more complicated example illustrating differential growth. We have an investment that will pay dividends of 1.12 units starting one year from now, growing at 12 % per year for the next 5 years. From then on, it will be growing at 8%. What is the present value of these dividends if the required return is 12%? Solution: first part, the present value for the first 6 years is a growing annuity
>

$\mathrm{part1}\u2254\mathrm{growingannuity}\left(1.12\,0.12\,0.12\,6\right)$

${\mathrm{part1}}{:=}{6.000000000}$
 (5) 
The fact that this is 6 times the present value of the first dividend is because the growth rate is equal to the required return. The second part, is a (deferred) growing perpetuity. Six years from now, the dividends will be
>

$\mathrm{div\_6}\u2254\mathrm{futurevalue}\left(1.12\,0.12\,5\right)$

${\mathrm{div\_6}}{:=}{1.973822685}$
 (6) 
So, the growing perpetuity, will start with dividends of
>

$\mathrm{div\_7}\u2254\mathrm{futurevalue}\left(\mathrm{div\_6}\,0.08\,1\right)$

${\mathrm{div\_7}}{:=}{2.131728500}$
 (7) 
Its value 6 years from now is
>

$\mathrm{part2\_6}\u2254\mathrm{growingperpetuity}\left(\mathrm{div\_7}\,0.12\,0.08\right)$

${\mathrm{part2\_6}}{:=}{53.29321250}$
 (8) 
Which has a present value of
>

$\mathrm{part2}\u2254\mathrm{presentvalue}\left(\mathrm{part2\_6}\,0.12\,6\right)$

${\mathrm{part2}}{:=}{27.00000000}$
 (9) 
Therefore the investment has a present value of
>

$\mathrm{part1}\+\mathrm{part2}$

33 units.